ZERO-DILATION INDEX OF S-n-MATRIX AND COMPANION MATRIX

Abstract

The zero-dilation index d(A) of a square matrix A is the largest k for which A is unitarily similar to a matrix of the form [GRAPHICS] , where 0(k) denotes the k-by-k zero matrix. In this paper, it is shown that if A is an S-n-matrix or an n-by-n companion matrix, then d (A) is at most. [n/2], the smallest integer greater than or equal to n/2. Those A's for which the upper bound is attained are also characterized. Among other things, it is shown that, for an odd n, the S-n-matrix A is such that d (A) = (n + 1) /2 if and only if A is unitarily similar to -A, and, for an even n, every n-by-n companion matrix A has d (A) equal to n/2.